W. Mathematica 8 stays running and does not evaluate

I am a user of Mathematica as a tool for calculation in Structural Mechanics. I had a problem recently in the integration of expressions a bit long, and honestly I do not understand what might be causing the slowdown.

I want to just evaluate this:

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This is for the Raileygh-Ritz method for rectangular plates. Then, i use the following code:

The symbolic integration of Integrate can take a very long time for complex functions, and in fact may never find an answer. If you know the values of all of the parameters, NIntegrate will run much faster and always will return an answer. You will need to assign all of the parameters beforehand, so that NIntegrate is just dealing with numbers, not symbolic expressions. I don't know if this is possible for you or not.

If you can't use NIntegrate, you might try Simplify on the symbolic expression before you use Integrate. Sometimes this helps.

Anyone have any solution or advise that could make my process more agile?

Thank you in advance!

Yep.

(1) We have a sub-forum "Computing and Tech/Math and Science software" down below. They're really good. Maybe this should be moved to that.

(2) No doubt in my mind even if the assignment was to find an analytic solution and I couldn't after some effort, I'd try and numerically solve it with just any old reasonable values for the necessary parameters, just to get a handle on it. In that case, use NIntegrate to see what's up.

(3) Re-cast the problem entirely in terms of a Mathematica problem and NIntegrate, try and solve it, if you run into problems, post a thread in the Science software forum describing where you're having problems.

Attached Files:

Part of the reason it is taking so long is that your integrand
((D[Wxy,{x,2}]+D[Wxy,{y,2}])^2 - 2*(1-v)*(D[Wxy,{x,2}]* D[Wxy,{y,2}]-(D[Wxy,x,y]^2)))
is 2 1/2 pages of expression and this
Expand[ ((D[Wxy,{x,2}]+D[Wxy,{y,2}])^2 - 2*(1-v)*(D[Wxy,{x,2}]* D[Wxy,{y,2}]-(D[Wxy,x,y]^2)))]
is about 40 pages of expression.

It is fortunate that in the Expanded version all the terms are fairly simple and have no denominators containing anything but small integer constants and Pi.

I had already waited a while and finally written "I suggest trying to Integrate the Expanded version and give it a day or three to see if you get an answer" when I was astonished to see:

That is on a fairly old slow machine. You need to verify this for yourself.

My thinking is that Integrate might be faster on an integrand with very simple terms, in spite of having lots of them, than it is on products of powers of D[] that it might be evaluating repeatedly, although it isn't supposed to be doing that.

So try it with and without Expand. Throw a dozen minutes or an hour at each version and see if you get an answer both ways.

I also got it to evaluate by simplifying the terms first and then forcing it to integrate term by term. The notebook is attached. I think it may be the same result as Bill Simpson's. here's the final result: